\newproblem{lay:1_4_13}{
  % Problem identification
	\begin{large}
	  \hspace{\fill}\newline
    \textbf{Lay, 1.4.13}
	\end{large}
	\\
  \ifthenelse{\boolean{identifyAuthor}}{\textit{Carlos Oscar Sorzano, Aug. 31st, 2013} \\}{}

  % Problem statement
	Let $\mathbf{u}=(0,4,4)$ and $A=\begin{pmatrix}3 & -5 \\ -2 & 6 \\ 1 & 1\end{pmatrix}$. Is $\mathbf{u}$ in the plane
	spanned by the columns of $A$? Why or why not?
	\begin{center}
		\includegraphics[scale=0.5]{Tema2/lay_1_4_13.jpg}
	\end{center}
}
{
  % Solution
	We need to solve the vector equation
	\begin{center}
		$\mathbf{u}=c_1\mathbf{a}_1+c_2\mathbf{a}_2$
	\end{center}
	or what is the same, the equation system represented by the augmented matrix below
	\begin{center}
		$\left(\begin{array}{rr|r} 3 & -5 & 0 \\ -2 & 6 & 4 \\ 1 & 1 & 4\end{array}\right) \sim
		 \left(\begin{array}{rr|r} 3 & -5 & 0 \\  0 & \frac{8}{3} & 4 \\ 0 & 0 & 0\end{array}\right)$
	\end{center}
	The system is compatible determinate, meaning that there exist $c_1$ and $c_2$ so that the vector equation is satisfied and, therefore,
	$\mathbf{u}$ belongs to the plane spanned by the columns of $A$.
}
\useproblem{lay:1_4_13}
\ifthenelse{\boolean{eachProblemInOnePage}}{\newpage}{}
